DINAMIČKI ODZIV PLANETARNOG PRENOSNIKA NA SPREŽNU PARAMETARSKU POBUDU

Unutrašnje dinamičke sile i pomeranja po predviđenim stepenima slobode u planetarnim prenosnicima posledica su, prvenstveno, promenljivih elastičnih deformacija zubaca u sprezi, vibracija delova prenosnika i odstupanja geometrije ozubljenja usled habanja zubaca i grešaka izrade. U radu se razmatra uticaj vremenski promenljive sprežne krutosti na moguća pomeranja i sile u kontaktima zubaca. Funkcije promene krutosti su korišćene kao nelinearni koeficijenti krutosti u sistemu jednačina za ekvivalentni zupčani prenosnik pri utvrđivanju prirode pomeranja i unutrašnjih dinamičkih sila. Rezultati numeričke analize sistema jednačina kojima je opisano dinamičko ponašanje su vremenski − promenljive funkcije pomeranja tačaka sistema i dinamičkih sila u spregama zubaca centralni zupčanik/satelit i satelit/venčanik.


Literatura
August, R., Kasuba, R., 1986, Torsional Vibrations and Dynamic  Internal dynamic forces and displacements along the anticipated degrees of freedom in planetary gear trains are a consequence of coupled teeth elastic deformations, vibrations of train parts and changes in gear geometries due to tooth wear and cutting errors.
This paper considers the influence of time-varying mesh stiffness on possible displacements and forces in tooth contacts.The timevarying mesh stiffness functions were useful as nonlinear coefficients of gear mesh stiffness in a system of differential equations for an equivalent gear train model in the determination of the nature of displacements and internal dynamic forces.The results of the numerical analysis of differential equations describing dynamic behaviour are time-varying functions of displacements of system points and dynamic forces in tooth meshes of the sun-planet and ring-planet meshes.

Introduction
Describing the dynamic behavior of a system includes, among other things, predicting the movement provided by the degrees of freedom and forces in the contact of gear teeth.The level and the character of the displacements and the forces in the sun gear/planet and planer/ring meshes were discussed in papers (August, Kasuba, 1986), (August, Evans, 1981), (Lin, Parker, 2002), (Kasuba, August, 1984), (Kahraman, 1994), (Ognjanović, 1985) on a mechanical-mathematical model that includes time-varying mesh stiffness values modeled as rectangular waveforms with different contact ratios.Such changes of mesh stiffness values are a rough approximation of actual changes that parametrically excite the system.
Changes in mesh stiffness dependent on the changes in the number of pairs of teeth in contact, on the tolerances in shape and on tooth wear are incorporated into the dynamic model of the planetary gear shown in this paper.

Mechanical-mathematical Model
To analyze the dynamic behavior of a planetary gear "a dynamic model of transmission should be defined first.Most authors who have dealt with this problem give the dynamic model as in Fig. 1.However, this model cannot show the connection with the drive gear and a working part having an impact on dynamics.Therefore, for the analysis of dynamic loads in the planetary gear, the complete transmission system (Fig. 2) was observed.This system includes a power source (plant), the load of machinery, gears, shafts and other damping sources.To eliminate the influence of the planet carrier on the dynamic characteristics of the meshes, an equivalent gear carrier was taken into consideration during the installation of a transmission system mechanical model.(Fig. 3).
The dynamics of the system is described by the differential equations from 4 to 10.

Numerical Analysis
The numerical analysis of this problem was done for a real planetary gear structure.

Conclusion
A model to simulate the dynamic behavior of single-stage planetary gears with straight-teeth gears is developed in this paper.Periodic excitation functions were used as vectors in accordance with the mesh stiffness and the parameters of static transmission errors assumed as known from the static-elastic analysis.
The calculation results are the functions of time changes of the system points displacements along the degrees of freedom and the forces in the S/P and P/R meshes.
The results obtained in the numerical example provide a basis for the following conclusions: -Forces in the S/P and P/R planetary set meshes are not constant but dependent on the elastic deformation of teeth of meshmesh stiffness and have an oscillatory character, -Force in the S/P mesh is more intense than the force in the P/R mesh, which is realistic given the contact ratio of meshing gears, -Sun gear displacement gives the ability to predict the dynamic force in the bearing.Key words: planetary gear train, dynamic forces, mesh stiffness, displacement, vibrations.